In particular, \(\diamondsuit (\mathcal {M},\mathbb {N}^\mathbb {N},\lnot \ni )\)  implies \(\diamondsuit (\mathrm {bb}^{\infty }(E^2),=^\infty )\) , which in turn implies \(\diamondsuit (\mathbb {N}^\mathbb {N},=^\infty )\) . In fact, \((\mathcal {M},\mathbb {N},\lnot \ni )\le _B(\mathbb {N}^\mathbb {N},=^\infty )\ast (\mathbb {N}^\mathbb {N},=^\infty )\)  (Proposition 4.14 in [1]}),Curiously, a result of Zapletal [2]} implies that \((\mathcal {M},\mathbb {N},\lnot \ni )\lnot \le _B(\mathbb {N}^\mathbb {N},=^\infty )\) . where the latter is the (Borel) sequential composition of the invariant \((\mathbb {N}^\mathbb {N},=^\infty )\)  with itself (cf. Definition 4.10 in [3]} or §4.1 of [1]}). As a result, in any canonical model, the principles \(\diamondsuit (\mathbb {N}^\mathbb {N},=^\infty )\) , \(\diamondsuit (\mathrm {bb}^{\infty }(E^2),=^\infty )\) , and \(\diamondsuit (\mathrm {non}(\mathcal {M}))\) , are equivalent.The question of whether apparently different \(\diamondsuit \)  principles which have the same evaluation are actually different remains, stubbornly, open.
